Harding. — Certain Decimal and Metrical Fallacies. 101 



due correlation between the two. There is a notable differ- 

 ence in this respect between the ancient system, universal in 

 the British Empire, and the modern French scale. While our 

 own system does not appear to be the best possible, this corre- 

 lation is duly observed ; in the French scheme it is neglected, 

 with the result that the nomenclature is unscientific and mis- 

 leading. It is the usual convention to break any long series 

 of figures into groups of three for the sole purpose of facilitat- 

 ing reading. The comma, usually employed for this purpose, 

 is no more a mark of grammatical punctuation than the period 

 when used as a decimal point. But French numeration is so 

 glaringly anti-arithmetical that it can be explained only on the 

 theory that in their inveterate habit of mistaking arbitrary 

 symbols for scientific facts the devisers of the scheme attached 

 some occult mathematic significance to the triple grouping. 

 In fixing the major radix the question arises at which stage to 

 abandon ten and substitute a multiple. Theoretically, as it is 

 easy to show, the second radix should be a power of ten which 

 is (1) a square number, and (2) the roots of which are also 

 squares till we reach the square of ten. Such numbers are 

 successively one hundred, ten thousand, and one hundred 

 millions. Either of these, radically subdivided, brings us in 

 the end to ten ; any other multiple of ten will yield as its root 

 number an interminable decimal. For the larger radix one 

 hundred is obviously too small ; multiplied by a million it is 

 inconveniently large. The point naturally indicated for the 

 break, therefore, is the myriad, in which case the ciphers 

 would properly be grouped not in triplets but in fours. By 

 this arrangement the numeration table would stand thus, the 

 square numbers being indicated by small capitals : — 



One ... ... ... ... 1 



First Series (Minor Radix). 



Ten ... ... ... ... 10 



Hundred (10'^) ... ... ... 100 



Thousand (10«) ... ... ... 1000 



Myriad (10^ = lOO'^) ... ... 1,0000 



Second Series (Major Radix). 



Decamyriad (10')... ... ... 10,0000 



Million (10'' = 1000^) ... ... 100,0000 



Milliard (10^) ... ... ... 1000,0000 



Billion (10« = 100^ = 1,0000-) ... 1,0000,0000 



Trillion (1,0000^^ = 100,0000'^) ... 1,0000,0000,0000 



Quadrillion (1,0000^ = 1,0000,0000^) 1,0000,0000,0000,0000 



In this scheme the first and second series are consistent 

 and complementary ; the powers of the minor radix are in- 



